X-ray Crystallography

About the Unit Cell

Crystals are three dimensional ordered structures than can be described
as a repetition of identical unit cells. The unit cell is made up of the
smallest possible volume that when repeated, is representative of the
entire crystal. The dimensions of a unit cell can be described with 3
edge lengths (a,b,c) and 3 angles (alpha, beta, gamma). The 3D location
of atoms within a unit cell can be listed as their x, y, z Cartesian
Coordinates. Space groups describe the symmetry of a unit cell, of which
there are 230 variations. In the molecular origami program, by clicking
on Use Crystallographic Info NOW, you can
experiment with all the different types of space groups.

Why are X-rays used ?

Using visible light, it will never be possible to see atoms under even
the most powerful of microscopes. In order for an object to be seen,
its size needs to be at least half the wavelength of the light being used
to see it. Since visible light has a wavelength much longer that the
distance between atoms it is useless to see molecules. In order to
see molecules it is necessary to use a form of electromagnetic
radiation with a wavelength on the order of bond lengths, such as
X-rays.

Why X-ray diffraction ?

Unfortunately, unlike with visible light, there is no known way to focus
x-rays with a lens. This causes an x-ray microscope to be
unfeasible unless someone finds a way of focusing x-rays. Until then it
is necessary to use crystals to diffract x-rays and create a diffraction
pattern which can be interpreted mathematically by a computer. This
turns the computer into a virtual lens, so it on a monitor we can
look at the structure of a molecule. Crystals are important
because by definition they have a repeated unit cell within them. The
x-ray diffraction from one unit cell would not be significant.
Fortunately, the repetition of unit cells within a crystal amplifies the
diffraction enough to give results that computers can turn into a picture.

Growing Crystals

To perform x-ray crystallography, it is necessary to grow crystals with
edges around 0.1-0.3 mm. Crystals are formed as the conditions in a
supersaturated solution slowly change. There are three degrees of
saturation in solution, and crystallographers take advantage of these
when growing crystals:

Unsaturated - where no crystals will form or grow.

Low supersaturated - where crystals will grow but no new ones
will form.

High supersaturated - where crystals will both form and grow.
One theory of crystal growth is to start by getting a few crystals to
grow in the highly supersaturated solution. Then the crystals are
exposed to a less saturated solution so they can grow. This is done
either by moving the crystals or changing the saturation of the solution.
For small molecules, growing large enough crystals is relatively simple.
By taking a supersaturated solution of solution and gradually changing
the conditions, crystals will begin to grow. If left undisturbed for a
few days ideally a few large crystals will grow.
Proteins are difficult to crystallize because of their complexity and the
fact that protein scientists are usually working with small amounts of
protein.There are various methods of growing protein crystals:

Vapor Diffusion -(Hanging Drop Method)

This is probably the most common ways of crystal growth. A drop of
protein solution is suspended over a reservoir containing buffer and
precipitant. Water diffuses from the drop to the solution leaving the
drop with optimal crystal growth conditions.

Batch crystallization

A saturated protein solution left in a sealed container to let the
crystals grow.

Microbatch crystallization

A drop of protein solution is put in inert oil and left to grow.
Here there probably is some diffusion of proteins into the oil, lowering
the saturation over time.

Macroseeding

A crystal is grown in a highly saturated solution and placed in a
less saturated one where only growth of the crystal will occur.

Microseeding

A few crystals are grown, then crushed, and put into a final solution
that combines them into a few nice crystals. This involves quite a bit
of experimentation with solutions' concentrations to get the desired
number of crystals.

Free interface diffusion

A container has levels of varying saturation. Crystals form
initially in the highly saturated part, but as the solution mixes, it
eventually only supports crystal growth.

Dialysis

Similar to the previous, but with a semipermeable membrane separating
the layers.

Proteins are crystallized on such a small scale that it is difficult to
reproduce concentrations. This makes crystallizing proteins almost more
of an art than a science, and sometimes multiple methods are tried before
crystals of the required size are grown.

When X-rays are beamed at the crystal, electrons diffract the X-rays,
which causes a diffraction pattern. Using the mathematical Fourier
transform these patterns can converted into electron
density maps. These maps show contour lines of electron density.
Since electrons more or less surround atoms uniformly, it is possible to
determine where atoms are located.
Unfortunately since hydrogen has only one electron, it is difficult to
map hydrogens.
To get a three dimensional picture, the crystal is rotated while a
computerized detector produces two dimensional electron density maps for
each angle of rotation. The third dimension comes from comparing the
rotation of the crystal with the series of images. Computer programs use
this method to come up with three dimensional spatial coordinates.